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ASSIGNMENT 3.13.1.a) [1 point] The time series of quarterly growth rates (in percent) of consumptionis shown below (for the complete sample period), as well as the histogram (plus anormal density with the same mean and variance) for the sample period 1970Q1-1999Q4. Skewness of the growth rates over this period is equal to −1.04, whilekurtosis is equal to 5.74. Both are very different from the normal values of 0 and3, such that the null hypothesis of normality is convincingly rejected when applyingthe Jarque-Bera test: The test statistic takes a value of 59.0, with p-value 0.000. 3

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Figure 1: Quarterly growth rates of US consumption, 1968Q4-2013Q4.

The first 12 (partial) autocorrelations are shown below, with an asterisk indicatingstatistical significance at the 5% level. We find significant autocorrelations at lages 1,2, 3, and 8, and significant partial autocorrelations at lags 1 and 3 (but not at lag 2).

the quarterly consumption growth rate, and φ4 (L) = 1 − φ1 L − φ2 L2 − φ3 L3 − φ4 L4 .Using the observations for t =1970Q1-1999Q4, we find the following least squaresestimates, with standard errors in parentheses: µ̂ = 0.860(0.119), φ̂1 = 0.196(0.093),φ̂2 = 0.144(0.093), φ̂3 = 0.208(0.093), and φ̂4 = −0.072(0.093). Note that the esti-mated coefficients for the second and fourth lag are not significantly different fromzero even at the 10% level. The R2 of the model is equal to 0.135.

The residuals from the AR(4) model have skewness equal to −0.89 and kurtosisequal to 5.50. Normality is clearly rejected as the Jarque-Bera statistics is equalto 47.0, with a p-value of 0.000. The (partial) autocorrelations of the residuals areshown below; for both, we find a significant value at k = 8. No significant (partial)autocorrelations appear for the squared residuals, such that heteroskedasticity doesnot seem to be an issue.

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The residuals are shown below. For both 1974Q4 and 1980Q2 we observe a largenegative residual, followed by a number of large positive residuals in subsequentquarters. This pattern is typical for the occurrence of an additive outlier in anAR-model with positive autoregressive coefficients (which is the case here). 3

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1970 1975 1980 1985 1990 1995

3.1.c) [1 point] Setting the observations for 1974Q4 and 1980Q2 equal to 0, we findthat skewness becomes equal to −0.27, while kurtosis is equal to 3.04. These valuesare such that the Jarque-Bera test statistic takes the value 1.44, with a p-valueequal to 0.49 such that normality cannot be rejected. This is also confirmed bythe histogram shown below, which is much more similar to the normal density thanbefore. Apparently, the two observations in 1974Q4 and 1980Q2 are responsible forthe strong signs of non-normality in the original time series. .9

with 1974Q4 and 1980Q2 set equal to 0.The autocorrelations and partial autocorrelations are shown above. We find quitesome differences compared to the estimates obtained before. In addition, the auto-correlation at lag 8 is no longer significant, while the partial autocorrelations noware significant for lags 1 to 4, providing a clearer pattern than the partial autocor-relations for the original series.

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3.1.d) [2.5 points] Based on the AR(4) model estimated in 3.1.b., we compute the t-statistics λ̂i (τ ) for testing whether an additive outlier (AO) or innovation outlier (IO)occurred at time τ , for τ = 1, . . . , T . These sequences of test statistics are shownin Figure 5. The largest absolute value of the test statistic for additive outliers isobtained for 1980Q2, with λ̂AO (τ ) = −5.45. The largest absolute value of the teststatistic for innovation outliers is also obtained for 1980Q2, with λ̂IO (τ ) = −4.83.For both statistics, the second largest (absolute) value is found for 1974Q4, withλ̂AO (τ ) = −3.54 and λ̂IO (τ ) = −2.96s. 3 3

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Figure 5: Values of the λ̂i (τ ) statistics for testing whether an AO or IO

occurred at time τ are shown on the left and right, respectively.

We use simulation to obtain the (finite-sample) distribution of the test statistics

λ̂AO = max1≤τ ≤T |λ̂AO (τ )| and λ̂IO = max1≤τ ≤T |λ̂IO (τ )| under the null hypothesisof no outliers, using 10,000 replications. The results of this exercise are shown inFigure 6. For both test statistics we find very small p-values, equal to 0.000. Wetherefore reject the null hypothesis of no outliers. For 1980Q2 (where the maximumis achieved, as noted above) the value of the λ̂AO (τ ) statistic is somewhat larger (inabsolute value) than the value of the λ̂IO (τ ) statistic. However, the difference is notvery large (while their distributions are very similar) so that it is not completelyobvious whether this observation is best characterized as an AO or as an IO. 1.2 1.2

Figure 6: Simulated distribution under the null hypothesis of λ̂i (τ ) statistics for testing for the presence of an AO or IO are shown on the left and right, respectively.

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3.1.e) [2.5 points] First, we estimate an AR(4) model in which the observations in1974Q4 and 1980Q2 are treated as AO’s, that is

φ4 (L)(yt − µ − δ1 d1974Q4,t − δ2 d1980Q2,t ) = εt

where φ4 (L) = 1−φ1 L−· · ·−φ4 L4 , and where d1974Q4,t and d1980Q2,t are dummy vari-ables for 1974Q4 and 1980Q2, respectively. Using the observations for t =1970Q1-1999Q4, we find the following least squares estimates, with standard errors in paren-theses: µ̂ = 0.903(0.088), φ̂1 = 0.201(0.092), φ̂2 = 0.292(0.089), φ̂3 = 0.223(0.092),φ̂4 = −0.300(0.090), δ̂1 = −2.336(0.508), and δ̂2 = −3.400(0.510). Note that allfour AR-coefficients are significantly different from zero at the 5% level in this case.We find negative values for δ̂1 and δ̂2 as expected, given that consumption growthexperienced large drops in both quarters. The estimates of δ1 and δ2 are significanteven at the 1% level, while they also are large in magnitude (compared to µ̂). 3

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1970 1975 1980 1985 1990 1995

The R2 of the model is equal to 0.426, which is more than three times as large asthe R2 for the AR(4) model estimated in 3.1.b (the residual variance is reducedby about 33% from 0.45 to 0.30). The residuals from the AR(4) model with AO’sare shown above. Comparing this with the ‘standard’ AR(4) model, the main dif-ferences (obviously) occur in the residuals for 1974Q4 and 1980Q2 and subsequentobservations. In the AR(4)-AO model, the residuals have skewness equal to −0.28and kurtosis equal to 3.09. Normality is not rejected, as the Jarque-Bera statisticsis equal to 1.56, with a p-value of 0.46. The (partial) autocorrelations of the resid-uals are shown above; for both, no significant values occur. This also applies to the(partial) autocorrelations of the squared residuals.

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lag are not significantly different from zero at the 10% level. The estimates of δ1and δ2 are negative and significant even at the 1% level, while they also are large inmagnitude (compared to µ̂.

The R2 of the model is equal to 0.347, which is two-and-a-half times as large as

the R2 for the AR(4) model estimated in 3.1.b (the residual variance is reduced byabout 25% from 0.45 to 0.34). The residuals from the AR(4) model with IO’s areshown below. Comparing this with the ‘standard’ AR(4) model, the main differences(obviously) occur in the residuals for 1974Q4 and 1980Q2, which are exactly equalto 0 in the IO-model. In the IO-model, the residuals have skewness equal to −0.24and kurtosis equal to 3.04. Normality is not rejected, as the Jarque-Bera statistics isequal to 1.17, with a p-value of 0.56. The (partial) autocorrelations of the residualsare shown in the table above; for both, no significant values occur. This also appliesto the (partial) autocorrelations of the squared residuals. 3

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1970 1975 1980 1985 1990 1995

Based on the above findings, it seems that the AR(4)-AO model should be preferred.This is based on the following arguments: • In the ‘standard’ AR(4) model, the patterns in the residuals around the obser- vations in 1974Q4 and 1980Q2 resemble additive outliers more than innovation outliers. • The AR(4)-AO model provides the best fit (in terms of R2 and significance of the AR-coefficients)

Finally, comparing Figure 7, which shows the fitted values and residuals from themodel with outliers, with Figure 3 shows that the largest residuals in 1974Q4 and1980Q2 obviously have disappeared, but some others remain; for example in 1973Q2.In fact, all residuals for the four quarters from 1973Q2-1974Q1 are relatively largeand negative. Also around other recession periods (1981, 1990-1), we observe rel-atively large negative residuals. It may be worthwhile to examine whether someof these observations also should be considered as outliers, or whether a nonlinearmodel (e.g. a threshold model) may be necessary to capture these.

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3.1.f) [1 point] We use the three estimated AR(4) models to obtain one-step aheadforecasts for the quarterly growth rates of consumption for the sample period 2000Q1-2013Q4.

The three series of forecasts are shown below, together with the actual values ofconsumption growth. We observe that the three forecasts are almost identical. TheMSPE’s are equal to 0.180 (AR(4)), 0.196 (AR(4)-AO) and 0.225 (AR(4)-IO). Hence,incorporating outliers in the model actually leads to less accurate forecasts. All threemodels produce biased forecasts, in the sense that the mean forecast errors of −0.171,−0.207 and −0.249 (for the AR(4), AR(4)-AO and AR(4)-IO models, respectively)are significantly different from 0 at the 5% significance level. From Figure 9, itappears that the mean reason for the negative bias are due to 2007-9, for whichall three models give forecasts that are much higher than the actual consumptiongrowth rate. 2.0

In sum, treating the observations in 1974Q4 and 1980Q2 as outliers is not usefulfrom a forecasting perspective. One final remark to qualify this statement is thatthe usefulness of the forecasts seems limited anyway: the variance of the actualquarterly growth rates over the forecast period is equal to 0.243, such that even theMSPE for the AR(4) forecasts is only about 25% smaller.